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 A259659 Expansion of phi(x^6) * f(-x)^3 / f(-x^3) in powers of x where phi(), f() are Ramanujan theta functions. 1
 1, -3, 0, 6, -3, 0, 1, -9, 0, 12, -3, 0, 6, -12, 0, 6, -3, 0, 7, -15, 0, 18, -6, 0, 0, -15, 0, 24, -6, 0, 6, -15, 0, 6, -9, 0, 7, -21, 0, 30, -3, 0, 6, -21, 0, 24, -6, 0, 12, -27, 0, 0, -9, 0, 12, -21, 0, 36, -6, 0, 1, -18, 0, 36, -12, 0, 6, -33, 0, 18, -9, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of phi(x^6) * b(x) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function. Expansion of q^(-3/4) * eta(q)^3 * eta(q^12)^2 / (eta(q^3) * eta(q^6)) in powers of q. Euler transform of period 12 sequence [ -3, -3, -2, -3, -3, -1, -3, -3, -2, -3, -3, -3, ...]. a(2*n + 1) = -3 * A227595(n). a(3*n + 1) = -3 * A259655(n). a(3*n + 2) = 0. EXAMPLE G.f. = 1 - 3*x + 6*x^3 - 3*x^4 + x^6 - 9*x^7 + 12*x^9 - 3*x^10 + 6*x^12 + ... G.f. = q^3 - 3*q^7 + 6*q^15 - 3*q^19 + q^27 - 9*q^31 + 12*q^39 - 3*q^43 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^6] QPochhammer[ x]^3 / QPochhammer[ x^3], {x, 0, n}]; eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-3/4)* eta[q]^3*eta[q^12]^2/(eta[q^3]*eta[q^6]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 17 2018 *) PROG (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^12 + A)^2 / (eta(x^3 + A) * eta(x^6 + A)), n))}; CROSSREFS Cf. A227595, A259655. Sequence in context: A226535 A005928 A113062 * A005889 A095712 A085753 Adjacent sequences:  A259656 A259657 A259658 * A259660 A259661 A259662 KEYWORD sign AUTHOR Michael Somos, Jul 02 2015 STATUS approved

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)